The phase of the Moon is determined by the relative positions of the Sun,
the Moon, and the Earth. If the Moon is between the Earth and the Sun,
then we have a new Moon with the shadowed side of the Moon facing the
Earth. If the Earth is between the Moon and the Sun, then we have a full
Moon.

For a quarter Moon, the triangle formed by the Sun, Earth, and Moon
has a right angle in it, with the Moon either preceding the Earth in its
orbital path or following the Earth. A good diagram/simulation to look at
can be found at: Virtual Reality Moon Phases.

There is one extra factor that determines how a phase of the Moon will
appear to a person standing on the Earth. This is the person's position on
the Earth. The phase shape and orientation is fixed for a given time, but
the orientation of the observer is not.

A person standing at the North
Pole would see an inverted Moon and phase compared to a person standing at
the South Pole. A person at the Equator would see a Moon phase rotated
90 degrees compared to a person at a pole. Observers at a pole of the
Earth would always see the crescent Moon with the points aligned nearly
vertically with the horizon.

Observers at the Equator would always see the
crescent Moon with the points aligned nearly horizontally with the horizon.
The appearance is complicated by the fact that the Moon's orbit is tilted
a few degree with respect to the Equator.

Since you are in Texas, which is fairly close to the Equator, you will
tend to see the bottom or top half of the quarter Moon lit. If you were on
holiday in Alaska, you would tend to see the quarter Moon oriented with
the left or right side lit. It would be the same Moon, but you would be
standing "sideways" in one place compared to the other.